Robotic mechanisms comprising only active joints are used widely in a number of application domains and the control of such mechanism is well understood. We will refer to such mechanisms as active mechanisms. Active mechanisms are particularly well suited for situations where the working volume of the mechanism is free of obstacles and environmental constraints on the motion of the mechanism. This is typically the case for applications of active robot mechanisms to manufacturing tasks, where the robot work-cell is specifically designed to suit the requirements of the robot mechanism. There are situations, however, where access to the working volume of a robotic mechanism can not be made unimpeded. In such situations access to the working volume is restricted by environmental constraints, such as small openings, tight passages and obstacles. With active mechanisms these constraints must be accommodated with time-consuming off-line programming to allow the mechanism to accomplish a given task without undesired contact with the environment. The success of task execution depends on the accuracy with which the programmer has captured the geometry of the environmental constraints and the accuracy with which complex coordinated motions of multiple joints of the robot mechanism are carried out. This approach is inflexible and error-prone, often leading to unintended or overly forceful interaction between the robot mechanism and the environment, potentially damaging the environment, the mechanism itself, or both.
An alternative approach to controlling robotic mechanisms in the presence of environmental constraints is to use robot mechanisms which include one or more passive joints. We will refer to such robot mechanisms as hybrid mechanisms. By ensuring that each link of a hybrid mechanism, which is constrained by an environmental constraint, is attached to one or more passive joints at the proximal end of the constrained link, such a hybrid mechanism can comply freely with the environmental constraints acting on the mechanism. (Throughout this document we will use the terms `proximal` and `distal` to mean `closer to` and `further from` the base of the robot, respectively. The base of the robot refers to the point where the robot is rigidly attached to the environment.) This arrangement ensures that neither the environment nor the mechanism itself will be damaged during task execution, which makes hybrid mechanisms the preferred solution for applications where access to the workspace is restricted and avoiding incidental damage to the environment is critical. However the use of passive joints significantly complicates control of the mechanism and so hybrid mechanisms are rarely used in practice. The difficulties in controlling hybrid mechanisms arise because the environmental constraints on the motion of the constrained elements and attached passive joints must be characterized and used to accurately predict the motion of the mechanism in response to a given displacement of active joints. Further, the control is complicated by the fact that the location where a given environmental constraint is acting on the mechanism may change as the mechanism moves. This requires that the control method be able to update the characterization of the environmental constraints on the motion of the mechanism at run-time.
We will define a mechanism to comprise a serial chain of two or more rigid links, connected by one or more joints. Each of the joints can be either active or passive. An active joint is equipped with an actuator (motor), which is capable of moving the joint, and an encoding device (encoder), which provides information about the position of the joint at any time. A mechanism consisting of only active joints will be referred to as an active mechanism. A mechanism comprising both active and passive joints will be referred to as a hybrid mechanism. An element of the mechanism will refer to either a joint or a link of the mechanism. The element whose motion relative to the workspace is being controlled will be referred to as the target element. Normally the target element will correspond to a tool or an instrument attached to the distal end of the mechanism, but could be, in general, any element of the mechanism. We will use the term sub-mechanism to mean a subset of a larger mechanism, the sub-mechanism comprising at least one element of the larger mechanism. We will define the pose of an element to be the position and orientation of the element, expressed with respect to a given (e.g., Cartesian) coordinate frame. We will distinguish between a desired pose of an element and an actual pose of an element. The desired pose of an element is the pose that the element is expected to attain as a result of the control action of a control method. The desired pose of the target element is input to the control method. The actual pose of an element is the element's current pose with respect to a given Cartesian coordinate system. We will define a pose difference between pose A and pose B of an element to be a function of the two poses. Normally, the result of evaluating the pose difference function will be the Euclidean distance between the positional parts of the two poses and a unit vector and angle corresponding to the finite rotation separating the orientational components of the two poses. However, other functions can be defined to represent the pose difference between two given poses of an element.
FIGS. 1 and 2 introduce the notational conventions used throughout this document and provide a brief overview of the state of the art in control of active mechanisms. FIG. 1a shows a simple mechanism consisting of 4 links (101, 104, 107, 110), three mechanical joints (102, 105, 108), and three actuators corresponding to the three mechanical joints (103, 106, 109, respectively). Each of the actuators comprises a motor (111), which delivers mechanical force or torque to move the joint, and a means of determining the angular or linear position of the joint (112), which enables closed-loop control of each of the joints. FIGS. 1b, 1c, 1d, and 1e detail the notational conventions for the four types of mechanical joints that will be used in this document. FIG. 1b shows a translational joint (121) and the corresponding symbolic representation (122) used in subsequent figures. FIG. 1c shows a rotary twist joint (141) and the corresponding symbolic representation (142). FIG. 1d shows an out-of-plane revolute joint (161) and the corresponding symbolic representation (162). Finally, FIG. 1e shows an in-plane revolute joint (181) and the corresponding symbolic representation (182).
FIG. 2 shows a flow diagram of a typical control method for Cartesian control of a target element of a robotic mechanism comprising only active joints. The method 200 begins by determining the positions of all joints of the mechanism (205). The position of a translational joint is expressed as a linear distance and the position of a rotary joint is expressed as an angular displacement. Standard mathematical techniques (known in the art as forward kinematic) are then used to compute the actual (current) pose of the target element (210). The actual pose of the target element is compared with the desired pose of the target element (215) and the control method is terminated (220) if the difference between the two poses is less than some predetermined amount, where the amount can be a vector quantity. The pose difference consists of a positional and an orientational component. If the pose difference is larger than the predetermined amount, the method continues by characterizing the effect of moving each of the joints on the resulting Cartesian displacement of the target element (225). This step is accomplished by analyzing the effect of moving each individual joint of the mechanism on the Cartesian displacement of the target element. For each joint j this mapping will be, in general, a nonlinear function of the joint positions of all joints appearing in the serial chain of the mechanism between the joint j and the target element. The combined nonlinear mapping, which includes the individual mappings for each of the joints, is referred to in the art as the Jacobian mapping. In general, the Jacobian mapping relates infinitesimal displacements of each of the joints of a mechanism to the resulting Cartesian displacement of an element of the mechanism. The Jacobian mapping is a nonlinear function of the joint positions of all joints of the mechanism and therefore takes on different numerical values for different configurations (joint position values) of the mechanism.
Referring again to FIG. 2, in step (230) of method 200 the desired motion of the target element is next characterized and expressed as a six-vector of positional and orientational change relative to the actual (current) pose of the mechanism. The incremental motion for each joint is then computed, such that the difference between the resulting actual pose of the target element and the desired pose of the target element is minimized (235). For most industrial and service robot mechanisms this step normally involves a straightforward evaluation of known equations, known in the art as inverse kinematic equations. For more complicated mechanisms, which do not admit closed-form inverse kinematic equations (such as robot mechanisms comprising more than six joints) this step may involve a nonlinear optimization. A number of techniques for carrying out nonlinear optimization computations are known in the art. A given iteration of the control method ends by moving each of the joints of the mechanism (240) by the incremental motions computed in step (235) above. The method then resumes at step (205) and continues until the pose difference in step (215) becomes less than the predetermined amount and the method terminates.
Very few examples of controlling hybrid passive/active mechanisms have been reported in the published literature. The prior art includes two examples of hybrid passive/active mechanisms which are being controlled in the presence of a single environmental constraint. Both examples arise in the context of laparoscopic surgery, where the constrained element is a laparoscope and the environmental constraint is the port of entry of the laparoscope into the patient. Due to the environmental constraint imposed by the port of entry into the patient, the motion of the laparoscope is limited to four degrees of freedom (d.o.f.) of motion: three orthogonal rotations about the port of entry and one translation d.o.f. along the long axis of the instrument.
Hurteau et. al. describe a robotic system for laparoscopic surgery where a robotic arm is connected to a laparoscope via a two-axis passive universal joint at the wrist of the robot. They use a manual teach pendant to independently control the translational motion of the robot's wrist, relying on the compliant passive linkage to position the laparoscope tip in azimuth, elevation and insertion depth. Each of the translational d.o.f. of the robot wrist is controlled manually by adjusting the corresponding knob or dial on the teach pendant.
Wang et. al. also use a robotic arm and a passive universal joint to position a laparoscope inside the patient, subject to the constraint imposed by the port of entry of the laparoscope into the patient. They add a driven instrument rotation stage to allow control of azimuth, elevation, rotation and insertion depth of the laparoscope. The details of their control method have not been made public, but several limitations of the method are apparent from the observed behavior of the system (as discussed later).
A less directly related reference on the subject of control of hybrid passive/active mechanisms has been disclosed by Jain et. al. in U.S. Pat. No. 5,377,310. In this work a robot manipulator comprising both active and passive joints is being controlled by estimating passive joint friction forces on passive joints and using these estimates to predict the dynamic behavior of the passive joints during high-speed motion of the manipulator due to the motion of the active joints. Dynamic parameters of all links and all actuators comprising the mechanism are assumed known.